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For a proton–electron system, the interaction potential is V = V0 ln(r/r0), where r0 is a constant. If Bohr’s model is assumed to hold, how does the radius rₙ depend on the principal quantum number n?

1. rₙ is directly proportional to n
2. rₙ is inversely proportional to n
3. rₙ is directly proportional to n²
4. rₙ is inversely proportional to n²

Correct answer: rₙ is directly proportional to n

Solution

With V = V0 ln(r/r0), force F = V0/r. Setting mv^2/r = V0/r gives mv^2 = V0, so v is constant. Bohr quantization mvr = n*hbar then gives r_n ∝ n.

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